Finite Group Databasebeta

← #20287802

Subgroups — #20287802

Order 6784.

380 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811358C211yesmaximal, characteristic, complemented
2#16811355C211yesmaximal, characteristic, complemented
3#16811013C211yesmaximal, characteristic, complemented
4#16811009C211yesmaximal, characteristic, complemented
5#16811008C211yesmaximal, characteristic, complemented
6#16810847C211yesmaximal, characteristic, complemented
7#16810841C211yesmaximal, characteristic, complemented
8#16810840C211yesmaximal, characteristic, complemented
9#16810819C211yesmaximal, characteristic, complemented
10#16810817C211yesmaximal, characteristic, complemented
11#16810752C211yesmaximal, characteristic, complemented
12#16810726C211yesmaximal, characteristic, complemented
13#16810721C211yesmaximal, characteristic, complemented
14#16810703C211yesmaximal, characteristic, complemented
15C53×#464C211yesmaximal, characteristic, complemented
16C2×#11719924C2211yescharacteristic
17C2×#11719924C2211yescharacteristic
18C2×#11719924C2211yescharacteristic
19C2×D424C2211yescharacteristic
20C2×D424C2211yescharacteristic
21C2×#11719922C2211yescharacteristic
22C2×#11719922C2211yescharacteristic
23#13133214C2211yescharacteristic
24#13133213C2211yescharacteristic
25#13133206C2211yescharacteristic
26#13133206C2211yescharacteristic
27#13133206C2211yescharacteristic
28#13133205—21no—
29#13133203—21no—
30#13133202—21no—
31#13133202—21no—
32#13133201—21no—
33#13133199—21no—
34Q8×Dic53C2211yescharacteristic
35C4×Dic106C2211yescharacteristic
36C4×D212—21no—
37C4×D212C2211yescharacteristic
38D4×Dic53C2211yescharacteristic
39#13133179C2211yescharacteristic
40#13133179C2211yescharacteristic
41#13133177—21no—
42#13133150C2211yescharacteristic
43#13133146C2211yescharacteristic
44#13133146C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133128C2211yescharacteristic
50#13133127C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×Q16×C53C2211yescharacteristic
53C2×SD16×C53C2211yescharacteristic
54#124×C53C2211yescharacteristic
55C4×Q8×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53C2311yescharacteristic
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64C22×Dic53—21no—
65C2×D212—21no—
66C2×D212—21no—
67C2×D212C2311yescharacteristic
68C2×D212C2311yescharacteristic
69C2×D212C2311yescharacteristic
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×Dic106D411yescharacteristic
74Q16×C53—41no—
75SD16×C53—41no—
76C2×C8×C53C2311yescharacteristic
77C2×C8×C53C2311yescharacteristic
78C4⋊C4×C53—21no—
79C4⋊C4×C53C2311yescharacteristic
80C22⋊C4×C53—21no—
81C22⋊C4×C53—21no—
82C42×C53—21no—
83C42×C53C2311yescharacteristic
84C22⋊Dic53—21no—
85#11719924—22no—
86#11719924—22no—
87#11719924—41no—
88#11719922—22no—
89#11719922—41no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53D411yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102#11719919D411yescharacteristic
103#11719919D411yescharacteristic
104C4×Dic53—21no—
105C4×Dic53C2311yescharacteristic
106C4×Dic53C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109D424—41no—
110D424—41no—
111C23×C53—21no—
112C22×D53—21no—
113C22×D53—21no—
114C22×D53—21no—
115Q8×C53—21no—
116Q8×C53—21no—
117Q8×C53—21no—
118Q8×C53—21no—
119D4×C53—21no—
120D4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2411yescharacteristic
127C22⋊D53—42no—
128C22⋊D53—81no—
129C2×Dic53—21no—
130C2×Dic53—81no—
131C2×Dic53C2×D411yescharacteristic
132C2×Dic53C2×D411yescharacteristic
133C2×Dic53C2×D411yescharacteristic
134C2×Dic53C2×D411yescharacteristic
135D212—21no—
136D212—21no—
137D212—21no—
138D212—21no—
139D212—21no—
140D212—21no—
141D212—42no—
142D212—81no—
143C4×D53—81no—
144C4×D53—81no—
145C4×D53—81no—
146Dic106—41no—
147C8×C53—21no—
148C8×C53—21no—
149#88322—21no—
150#88322#4112yes—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53C22×D411yescharacteristic
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53C2×#4111yescharacteristic
170C4×C53H(5,𝔽2)11yescharacteristic
171Dic53—21no—
172Dic53—21no—
173Dic53—21no—
174Dic53—21no—
175Dic53—41no—
176#3403—531nomaximal, Sylow
177C2×C53—81no—
178C2×C53#55011yescharacteristic
179C2×C53#57511yescharacteristic
180C2×C53#57611yescharacteristic
181D53—81no—
182D53—81no—
183D53—81no—
184#559—531no—
185#559—531no—
186#496—531no—
187#496—531no—
188#495—531no—
189#464—531no—
190#464D5311yescharacteristic, complemented
191#463—531no—
192#463—531no—
193#459—531no—
194#459—531no—
195#446—531no—
196#440—531no—
197#440—531no—
198C4×D8—531no—
199C53#340311yescharacteristic, complemented, Sylow
200C2×Q16C2×D5311yescharacteristic
201C2×SD16—531no—
202C2×SD16—531no—
203C2×SD16—531no—
204C2×SD16C2×D5311yescharacteristic
205C2×D8—531no—
206C2×D8—531no—
207C2×D8—531no—
208C2×D8—531no—
209#127—531no—
210#127—531no—
211#124—531no—
212#124—531no—
213#124—531no—
214#124—1061no—
215#124—1061no—
216#124C2×D5311yescharacteristic
217#123—1061no—
218#123—1061no—
219#123—1061no—
220#123—1061no—
221C4×Q8—531no—
222C4×Q8—531no—
223C4×Q8C2×D5311yescharacteristic
224C4×D4—531no—
225C4×D4—531no—
226C4×D4—531no—
227C4×D4—531no—
228C4×D4—1061no—
229C4×D4—1061no—
230#107—531no—
231C4⋊C8—531no—
232C4⋊C8—531no—
233C4⋊C8C2×D5311yescharacteristic
234#103—531no—
235#103C2×D5311yescharacteristic
236#102—531no—
237#102—531no—
238#102—531no—
239#102C2×D5311yescharacteristic
240#97—531no—
241#97—531no—
242C4×C8—531no—
243C2×Q8—531no—
244C2×Q8C22×D5311yescharacteristic
245C2×Q8C22×D5311yescharacteristic
246C2×D4—531no—
247C2×D4—531no—
248C2×D4—531no—
249C2×D4—1061no—
250C2×D4—1061no—
251C2×D4—1061no—
252C2×D4—1061no—
253C2×D4C22×D5311yescharacteristic
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258Q16—41no—
259SD16—41no—
260SD16—1062no—
261SD16—1062no—
262SD16—2121no—
263D8—1062no—
264D8—2121no—
265D8—2121no—
266D8—2121no—
267C2×C8—531no—
268C2×C8—531no—
269C2×C8C22×D5311yescharacteristic
270C2×C8C22×D5311yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—531no—
273C4⋊C4—531no—
274C4⋊C4—531no—
275C4⋊C4—531no—
276C4⋊C4—1061no—
277C4⋊C4C22×D5311yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C42—21no—
291C42—531no—
292C42—531no—
293C42—1061no—
294C42C22×D5311yescharacteristic
295C23—21no—
296C23—1061no—
297C23—1061no—
298C23—1061no—
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—2121no—
304D4—21no—
305D4—21no—
306D4—1061no—
307D4—1061no—
308D4—1061no—
309D4—1061no—
310D4—1061no—
311D4—1061no—
312D4—4241no—
313D4—4241no—
314D4—2122no—
315D4—2122no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—531no—
320C2×C4—531no—
321C2×C4—531no—
322C2×C4—531no—
323C2×C4—1061no—
324C2×C4—4241no—
325C2×C4—4241no—
326C2×C4—4241no—
327C2×C4—4241no—
328C2×C4D4×D5311yescharacteristic
329C2×C4D4×D5311yescharacteristic
330C2×C4C23×D5311yescharacteristic
331C8—21no—
332C8—21no—
333C8—532no—
334C8—1061no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—2121no—
339C22—2121no—
340C22—2121no—
341C22—2121no—
342C22—2121no—
343C22—2121no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22C2×D4×D5311yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—1061no—
354C4—1061no—
355C4—1061no—
356C4—1061no—
357C4—2121no—
358C4#41×D5311yescharacteristic
359C4#1313326211yescharacteristic
360C2—81no—
361C2—4241no—
362C2—4241no—
363C2—4241no—
364C2#1681129111yescharacteristic
365C2#1681142211yescharacteristic
366C2#1681142511yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#464C2
socleC22×C53C22×D4
hypercenter#464D53
hypocenterC53#3403
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #16811358C2semidirect
2. #16811355C2semidirect
3. #16811013C2semidirect
4. #16811009C2semidirect
5. #16811008C2semidirect
6. #16810847C2semidirect
7. #16810841C2semidirect
8. #16810840C2semidirect
9. #16810819C2semidirect
10. #16810817C2semidirect
11. #16810752C2semidirect
12. #16810726C2semidirect
13. #16810721C2semidirect
14. #16810703C2semidirect
15. C53×#464C2semidirect
190. #464D53semidirect
199. C53#3403semidirect